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GATE 2019 CE (CE2) – Question 45

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

Find the inverse of $$\begin{bmatrix}2&3&4\\4&3&1\\1&2&4\end{bmatrix}.$$

  1. $$\begin{bmatrix}10&-4&-9\\-15&4&14\\5&-1&-6\end{bmatrix}$$
  2. $$\begin{bmatrix}-10&4&9\\15&-4&-14\\-5&1&6\end{bmatrix}$$
  3. $$\begin{bmatrix}-2&4/5&9/5\\3&-4/5&-14/5\\-1&1/5&6/5\end{bmatrix}$$
  4. $$\begin{bmatrix}2&-4/5&-9/5\\-3&4/5&14/5\\1&-1/5&-6/5\end{bmatrix}$$

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Correct answer: (C) $$\begin{bmatrix}-2&4/5&9/5\\3&-4/5&-14/5\\-1&1/5&6/5\end{bmatrix}$$

Explanation

The determinant is $2(12-2)-3(16-1)+4(8-3)=-5$. The adjugate is $$\begin{bmatrix}10&-4&-9\\-15&4&14\\5&-1&-6\end{bmatrix}.$$ Divide each entry by −5 to obtain **C**. Multiplying it by the original matrix gives the identity.